KEPLER Exoplanets Database

Observed exoplanet's periods VS. theoretical period calculated with stimated star mass and exoplanet mass.

Source: https://data.world/markmarkoh/kepler-confirmed-planets/workspace/project-summary?agentid=markmarkoh&datasetid=kepler-confirmed-planets NASA Exoplanet archive: https://exoplanetarchive.ipac.caltech.edu/docs/data.html

@Author: Javier Cebrián

@Attention: In this file there are Plotly (rendered with HTML) plots and equations. If you are viewing it with github, please enable external view with nbviewer

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import plotly.express as px
import plotly.graph_objects as go
import plotly.offline as pyo

sns.set()

planetsDf=pd.read_csv('../planets.csv',delimiter=',')

Correlation Obital Period [days] & Orbit Semi-Major Axis [AU]

Orbit Period and Semi-major Axis is related with the following equation:

Where a=semi-major axis , T=period and μ=MG with M=Mass of star and G=Gravity cte

$ T ∝ a^{3/2} $

planetsDf[['pl_orbsmax','pl_orbper']].corr()

	pl_orbsmax	pl_orbper
pl_orbsmax	1.000000	0.815671
pl_orbper	0.815671	1.000000

Whith scatter representacion can be observed that correlation value is affected by Oph 11 exoplanet outlier.

fig = px.scatter(x=planetsDf['pl_orbsmax'], y=planetsDf['pl_orbper'],title="Exoplanets: Orbital axis vs. period",width=800, height=320)
fig.update_xaxes(range=(0,300), title='Semi-major axis [AU]')
fig.update_yaxes(title='Period [days]')
fig.show()

Oph 11 is an outlier

planetsDf=planetsDf.drop(index=3183)

planetsDf[['pl_orbsmax','pl_orbper']].corr()

	pl_orbsmax	pl_orbper
pl_orbsmax	1.000000	0.965909
pl_orbper	0.965909	1.000000

fig = px.scatter(planetsDf,x='pl_orbsmax', y='pl_orbper',title="Exoplanets: Orbital axis vs. period",width=800, height=320)
fig.update_xaxes(range=(0,200), title='Semi-major axis [AU]')
fig.update_yaxes(title='Period [days]')
fig.show()

Error theory aproximation for planets with stars of known mass:

Applying the theory described at the beginning here is calculated the approximate result of the extrasolar planet's orbital period.

$ T \approx \sqrt{\frac{a^3}{ \frac{GM}{4 \pi^2}}}\ $

With a=semi-major axis , M=Mass of star and G=Gravity

a = 'pl_orbsmax' is the semi-major axis.

M = 'st_mass' is the mass star in solar masses.

4434075.792 is a factor to use solar masses and UA units

planetsDf=planetsDf[planetsDf['st_mass'].notnull()] #Erase data without star mass stimation
theorPeriod=np.power(planetsDf['pl_orbsmax'],3/2)/np.power(6.674E-11*(planetsDf['st_mass'])*4434075.792/(4*(np.power(np.pi,2))),1/2)
theorPeriod[planetsDf['pl_orbper'].isnull()]=np.nan
planetsDf['theorPeriod']=theorPeriod

data0 = go.Scatter(x=planetsDf['pl_orbsmax'], y=planetsDf['pl_orbper'], mode = 'markers', name='Measures')
data1 = go.Scatter(x=planetsDf['pl_orbsmax'], y=planetsDf['theorPeriod'], mode = 'markers', name='Theory')
data = [data0, data1]
layout = go.Layout(title='Observation Vs. Aproximation 1')
fig = go.Figure(data= data, layout = layout)
fig.update_xaxes(range=(0,200), title='Semi-major axis [AU]')
fig.update_yaxes(title='Period [days]')
fig.show()
#pyo.plot(fig, filename = 'line_chart.html')

Mean squared error:

((planetsDf['pl_orbper']-planetsDf['theorPeriod'])**2).mean()

721093.6716320177

More exactly:

$ T = \sqrt{\frac{a^3}{ \frac{G(M+m)}{4 \pi^2}}}\ $

With a=semi-major axis , M=Mass of star, m=mass of the planet and G=Gravity

a = 'pl_orbsmax' is the semi-major axis.

M = 'st_mass' is the mass star in solar masses.

4434075.792 is a factor to use solar masses and UA units

m = 'pl_bmassj' is the planet mass in Jupiter masses (Jupiter has $9.55·10^{-4}$ solar masses)

planetsDf=planetsDf[planetsDf['pl_bmassj'].notnull()] #Erase data without planet mass stimation
theorPeriod=np.power(planetsDf['pl_orbsmax'],3/2)/np.power(6.674E-11*(planetsDf['st_mass']+planetsDf['pl_bmassj']*9.55*10**-4)*4434075.792/(4*(np.power(np.pi,2))),1/2)
theorPeriod[planetsDf['pl_orbper'].isnull()]=np.nan
planetsDf['theorPeriod']=theorPeriod

data0 = go.Scatter(x=planetsDf['pl_orbsmax'], y=planetsDf['pl_orbper'], mode = 'markers', name='Measures')
data1 = go.Scatter(x=planetsDf['pl_orbsmax'], y=planetsDf['theorPeriod'], mode = 'markers', name='Theory')
data = [data0, data1]
layout = go.Layout(title='Observation Vs. Aproximation 2')
fig = go.Figure(data= data, layout = layout)
fig.update_xaxes(range=(0,100), title='Semi-major axis [AU]')
fig.update_yaxes(title='Period [days]')
fig.show()
#pyo.plot(fig, filename = 'line_chart.html')

Mean squared error:

((planetsDf['pl_orbper']-planetsDf['theorPeriod'])**2).mean()

30708.99206598928

Conclusion:

Correlation shows relation between observed period and orbital axis. The observations conform to the theoretical calculation:

$ T = \sqrt{\frac{a^3}{ \frac{G(M+m)}{4 \pi^2}}}\ $

With a=semi-major axis , M=Mass of star, m=mass of the planet and G=Gravity

The relation measured is causal.